You won the lottery! Your prize consists of the following annual payments: You will receive a payment of $0.05 immediately, which will then double each year for 30 years. The APR is 18%, compounded monthly. What is the present value of these cash flows?

We are tasked with finding the present value of a sequence of cash flows where the initial payment is $0.05 and it doubles each year for 30 years. The APR (Annual Percentage Rate) is 18%, compounded monthly.

### Step 1: Identify the cash flows
The cash flow at the end of each year is as follows:
- Year 0: $0.05 (immediate payment)
- Year 1: $0.10 (doubles from Year 0)
- Year 2: $0.20 (doubles from Year 1)
- And so on...

The general formula for the cash flow in year $n$ is:

\[
\text{Cash Flow in Year } n = 0.05 \times 2^n
\]

### Step 2: Adjust for the APR compounded monthly
The APR is 18%, compounded monthly. The monthly interest rate $i$ can be calculated as:

\[
i = \frac{18\%}{12} = 1.5\% \text{ per month}
\]

To compute the equivalent annual discount rate, we convert the monthly rate to an annual effective rate. The formula for converting a nominal monthly interest rate to an effective annual rate (EAR) is:

\[
\text{EAR} = \left(1 + \frac{\text{APR}}{12}\right)^{12} - 1
\]

Thus:

\[
\text{EAR} = (1 + 0.015)^{12} - 1
\]

### Step 3: Present value of each cash flow
The present value of a future cash flow is given by:

\[
\text{PV} = \frac{\text{Future Cash Flow}}{(1 + \text{EAR})^n}
\]

For each year $n$, we will compute the present value of the cash flows and sum them up.

Let me compute these values for you.The present value of the cash flows is approximately **$627,570.26**. The effective annual rate (EAR) is **19.56%**.

Would you like more details on any part of the calculation, or do you have any further questions?

Here are 5 related questions for further exploration:
1. How does the present value change if the APR is compounded quarterly instead of monthly?
2. What would be the present value if the payments only double every two years?
3. How would the present value change if the payment sequence started from a higher initial amount, like $0.10?
4. How does changing the time period from 30 years to 40 years affect the present value?
5. What if the APR were 10% instead of 18%?

**Tip:** The compounding frequency significantly impacts the present value of future cash flows; monthly compounding leads to higher effective rates than annual compounding.

Calculate the present value of a lottery prize with annual payments doubling each year for 30 years, starting at $0.05. Understand the impact of an 18% APR compounded monthly on the present value of exponentially growing cash flows. Includes key concepts like time value of money, present value calculation, and effective annual rate.

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